In this paper, we study local rings of small Hilbert-Kunz multiplicity. In particular, we prove that an unmixed local ring of Hilbert-Kunz multiplicity one is regular and classify two-dimensional Cohen-Macaulay local rings whose Hilbert-Kunz multiplicity is 2 or less. Also, we investigate the inequality between the multiplicity and the colength of the tight closure of parameter ideals inverse to the usual inequality between multiplicity and colength.

In this paper we study Ulrich ideals of and Ulrich modules over Cohen-Macaulay local rings from various points of view. We determine the structure of minimal free resolutions of Ulrich modules and their associated graded modules, and classify Ulrich ideals of numerical semigroup rings and rings of finite CM-representation type.

In this paper, we introduce the notion of p g -ideals and p g -cycles, which inherits nice properties of integrally closed ideals on rational singularities. As an application, we prove an existence of good ideals for two-dimensional Gorenstein normal local rings. Moreover, we classify all Ulrich ideals for two-dimensional simple elliptic singularities.

Abstract. In this paper, we investigate the lower bound sHK(p, d) of HilbertKunz multiplicities for non-regular unmixed local rings of Krull dimension d containing a field of characteristic p > 0. Especially, we focus on threedimensional local rings. In fact, as a main result, we will prove that sHK(p, 3) = 4/3 and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity 4/3 is isomorphic to the non-degenerate quadric hypersurfaceFurthermore, we pose a generalization of the main theorem to the case of dim A ≥ 4 as a conjecture, and show that it is also true in case dim A = 4 using the similar method as in the proof of the main theorem.

Abstract. In this paper, we prove that for any ideal / of dimension one H}(M) is /-cofinite for all i and for any finite A-module M. Furthermore, for any ideal / over any regular local ring A, we investigate the relationship between /-cofiniteness and vanishing for local cohomology modules H}(M). §1. Main TheoremLet A be a Noetherian ring. For an ideal / of A and an yl-module Tkf, we setIn general, a functor H}(-) is the i right derived functor of the functor Γ / (-) = lim fc Hom(A// fe , -). In this note, refining their proof, we prove the following theorem. Let M be a finite A-module. Then for any finite A-module N such that Supp^(iV) C V(I), we have Ext^(7V, Hj-(M)) is of finite type for all i, j.In order to prove Theorem 1.1, we need fundamental two lemmas. (1) Ext^(τ4//, V) is a finite A-module for all t < p.(2) Ext^(A/\/7 5 ^0 is a finite A-module for all t < p.